|
| | QGaussOneOverR (const unsigned int n, const Point< dim > &singularity, const bool factor_out_singular_weight=false) |
| |
| | QGaussOneOverR (const unsigned int n, const unsigned int vertex_index, const bool factor_out_singular_weight=false) |
| |
| | QGaussOneOverR (const unsigned int n, const Point< 2 > &singularity, const bool factor_out_singularity) |
| |
| | QGaussOneOverR (const unsigned int n, const unsigned int vertex_index, const bool factor_out_singularity) |
| |
| bool | operator== (const Quadrature< dim > &p) const |
| |
| void | initialize (const std::vector< Point< dim > > &points, const std::vector< double > &weights) |
| |
| unsigned int | size () const |
| |
| const Point< dim > & | point (const unsigned int i) const |
| |
| const std::vector< Point< dim > > & | get_points () const |
| |
| double | weight (const unsigned int i) const |
| |
| const std::vector< double > & | get_weights () const |
| |
| std::size_t | memory_consumption () const |
| |
| template<class Archive > |
| void | serialize (Archive &ar, const unsigned int version) |
| |
| bool | is_tensor_product () const |
| |
| const std::array< Quadrature< 1 >, dim > & | get_tensor_basis () const |
| |
template<
int dim>
class QGaussOneOverR< dim >
A class for Gauss quadrature with \(1/R\) weighting function. This formula can be used to integrate \(1/R \ f(x)\) on the reference element \([0,1]^2\), where \(f\) is a smooth function without singularities, and \(R\) is the distance from the point \(x\) to the vertex \(\xi\), given at construction time by specifying its index. Notice that this distance is evaluated in the reference element.
This quadrature formula is obtained from two QGauss quadrature formulas, upon transforming them into polar coordinate system centered at the singularity, and then again into another reference element. This allows for the singularity to be cancelled by part of the Jacobian of the transformation, which contains \(R\). In practice the reference element is transformed into a triangle by collapsing one of the sides adjacent to the singularity. The Jacobian of this transformation contains \(R\), which is removed before scaling the original quadrature, and this process is repeated for the next half element.
Upon construction it is possible to specify whether we want the singularity removed, or not. In other words, this quadrature can be used to integrate \(g(x) = 1/R\ f(x)\), or simply \(f(x)\), with the \(1/R\) factor already included in the quadrature weights.
Definition at line 300 of file quadrature_lib.h.
This constructor takes three arguments: the order of the Gauss formula, the point of the reference element in which the singularity is located, and whether we include the weighting singular function inside the quadrature, or we leave it in the user function to be integrated.
Traditionally, quadrature formulas include their weighting function, and the last argument is set to false by default. There are cases, however, where this is undesirable (for example when you only know that your singularity has the same order of 1/R, but cannot be written exactly in this way).
In other words, you can use this function in either of the following way, obtaining the same result:
for(unsigned int i=0; i<singular_quad.size(); ++i)
integral += f(singular_quad.point(i))*singular_quad.weight(i);
for(unsigned int i=0; i<singular_quad.size(); ++i) {
double R = (singular_quad_noR.point(i)-cell->vertex(vertex_id)).
norm();
integral += f(singular_quad_noR.point(i))*singular_quad_noR.weight(i)/R;
}
double norm(const FEValuesBase< dim > &fe, const ArrayView< const std::vector< Tensor< 1, dim > > > &Du)
The constructor takes three arguments: the order of the Gauss formula, the index of the vertex where the singularity is located, and whether we include the weighting singular function inside the quadrature, or we leave it in the user function to be integrated. Notice that this is a specialized version of the previous constructor which works only for the vertices of the quadrilateral.
Traditionally, quadrature formulas include their weighting function, and the last argument is set to false by default. There are cases, however, where this is undesirable (for example when you only know that your singularity has the same order of 1/R, but cannot be written exactly in this way).
In other words, you can use this function in either of the following way, obtaining the same result:
for(unsigned int i=0; i<singular_quad.size(); ++i)
integral += f(singular_quad.point(i))*singular_quad.weight(i);
for(unsigned int i=0; i<singular_quad.size(); ++i) {
double R = (singular_quad_noR.point(i)-cell->vertex(vertex_id)).
norm();
integral += f(singular_quad_noR.point(i))*singular_quad_noR.weight(i)/R;
}
1.9.4